Examples

Of ∞\infty-group actions in an ∞\infty-topos

Let H\mathbf{H} be an (∞,1)-topos and let G∈Grp(H)G \in Grp(\mathbf{H}) be an ∞-group in H\mathbf{H}.

The following lists some fundamental classes of examples of ∞\infty-actions of GG, and of other canonical ∞\infty-groups. By the discussion above these actions may be given by the classifying morphisms.

is the configuration space of fields on Σ\Sigma modulo automorphisms (diffeomorphisms, in smooth cohesion) of Σ\Sigma. This is the configuration space of “generally covariant” field theory on Σ\Sigma.

Semidirect product groups

Let G,A∈Grp(H)G, A \in Grp(\mathbf{H}) be 0-truncated group objects and let ρ\rho be an action of GG on AA by group homomorphisms. This is equivalently an action of GG on BA\mathbf{B}A, hence a fiber sequence

which is still Cartesian, by this proposition. Use that the bottom left object here is equivalently B≃∑BB*(*)B \simeq \underset{B}{\sum} B^\ast (\ast) and form the pasting with the naturality square of the (∑B⊣B*)(\underset{B}{\sum}\dashv B^\ast)-counit.

By this proposition also this naturality square is Cartesian. Hence by the pasting law the total rectangle is Cartesian. This exhibits the AutH(L)\mathbf{Aut}_{\mathbf{H}}(L)-action on X=∑BLX = \underset{B}{\sum} L.

Remark

Stated more intuitively, prop. 4 says that sliced automorphisms of the form

Co-Discretization of Actions

Proposition

Given an ∞-groupGG in H\mathbf{H} and a GG-action, def. 2, on some XX, then ♯nG\sharp_n G is itself canonically an ∞\infty-group equipped with a canonically induced action on ♯nX\sharp_n X such that the projection X→♯nXX \to \sharp_n X carries the structure of a homomorphism of GG-actions.

Proof

Observe that ♯n\sharp_n preserves products, since ♯\sharp does (being a right adjoint) and by this proposition. Now use that the homotopy quotientV/GV/G is the realization of the simplicial object(V/G)•=G×•×V(V/G)_\bullet = G^{\times_{\bullet}} \times V. So applying ♯n\sharp_n to this yields a simplicial object ((♯nV)/(♯nG))•=(♯nG)×•×(♯nV)((\sharp_n V)/(\sharp_n G))_\bullet = (\sharp_n G)^{\times_{\bullet}} \times (\sharp_n V) which exhibits the desired action.

Proof

Generally, let A:B→TypeA:B\to Type be any dependent type family (speaking homotopy type theory). We claim that there is an induced family A♯n:♯n+1B→TypeA^{\sharp_n} : \sharp_{n+1} B \to Type such that A♯n(ηn+1(b))=♯n(A(b))A^{\sharp_n}(\eta_{n+1}(b)) = \sharp_n (A(b)) for any b:Bb:B, where ηn+1:B→♯n+1B\eta_{n+1} : B \to \sharp_{n+1} B is the inclusion. Applying this when A→BA \to B is V/G→BGV/G \to \mathbf{B}G and when bb is (necessarily) the basepoint of BG\mathbf{B}G gives the desired action on the desired type.

First of all, we have the composite B→AType→♯Type♯B \xrightarrow{A} Type \xrightarrow{\sharp} Type_{\sharp}, where Type♯=∑X:Typeis♯(X)Type_{\sharp} = \sum_{X:Type} is\sharp(X). Since Type♯Type_{\sharp} is itself ♯\sharp (since ♯\sharp is lex), this factors through ♯B\sharp B, giving a type family A♯:♯B→Type♯A^\sharp : \sharp B \to Type_{\sharp} such that A♯(η(b))=♯(A(b))A^{\sharp}(\eta(b)) = \sharp (A(b)) for any b:Bb:B, where η:B→♯B\eta:B\to \sharp B is the unit of ♯\sharp.

Now fix y:♯By:\sharp B and x:A♯(y)x:A^\sharp(y). For any b:Bb:B and p:η(b)=yp:\eta(b)=y, we can define the type ‖∑(a:A(b))p*(η(a))=x‖n{\big\Vert \sum_{(a:A(b))} p_\ast (\eta(a)) = x\big\Vert}_n. This is an nn-type, and since the type oftruncated typesn-Typen\text{-}Type is an (n+1)(n+1)-type, as a function of (b,p):∑b:Bη(b)=y(b,p) : \sum_{b:B} \eta(b)=y, this construction factors through ‖∑b:Bη(b)=y‖n+1\big\Vert \sum_{b:B} \eta(b)=y\big\Vert_{n+1}. Thus, for y:♯By:\sharp B and x:A♯(y)x:A^\sharp(y) and ξ:‖∑(b:B)η(b)=y‖n+1\xi : {\big\Vert \sum_{(b:B)} \eta(b)=y\big\Vert}_{n+1} we have a type P(y,x,ξ)P(y,x,\xi), such that